Hilbert-type Inequalities Research Articles

The Best Constants in Multidimensional Hilbert-Type Inequalities Involving Some Weighted Means Operators

In this paper, we establish several multidimensional Hilbert-type inequalities with a homogeneous kernel, involving the weighted geometric and harmonic mean operators in the integral case. The general results are derived for the case of non-conjugate parameters. A special emphasis is dedicated to determining conditions under which the obtained inequalities include the best possible constants on their right-hand sides, which can be established after reduction to the conjugate case. As an application, we consider some particular examples and compare our results with the previously known from the literature.

On a multidimensional version of the Hilbert-type inequality in the whole plane

Abstract The main objective of this paper is a study of some new multidimensional Hilbert-type inequalities with a general homogeneous kernel in the whole plane. We derive a pair of equivalent inequalities, and we also establish the conditions under which the constant factors included in the obtained inequalities are the best possible. Some applications in particular settings are also considered. MSC:26D15.

On reverse Hilbert-type inequalities

By introducing two pairs of conjugate exponents and estimating the weight coefficients, we establish reverse versions of Hilbert-type inequalities, as described by Jin (J. Math. Anal. Appl. 340:932-942, 2008), and we prove that the constant factors are the best possible. As applications, some particular results are considered.

Extremal functions in de Branges and Euclidean spaces

In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on RN. These extremal functions minimize the L1(RN,|x|2ν+2−Ndx)-distance to the original function, where ν>−1 is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function Fλ(x)=e−λ|x|, where λ>0, in the general framework of de Branges spaces of entire functions. We then specialize the construction to a particular family of homogeneous de Branges spaces to approach the multidimensional Euclidean case. Finally, we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter λ>0 against suitable measures. Applications of the results presented here include multidimensional versions of Hilbert-type inequalities, extremal one-sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one-sided approximations by polynomials for a class of functions on the sphere SN−1 with an axis of symmetry.

On a more accurate class of discrete Hilbert-type inequalities

Motivated by some known results regarding a particular set of non-homogeneous kernels, in this article we study a more general class of discrete Hilbert-type inequalities. We derive a more accurate form for this class of inequalities, based on the application of the Hermite–Hadamard inequality.

Half-discrete Hilbert-type inequalities involving differential operators

Abstract Motivated by some recent results, in this paper we derive several half-discrete Hilbert-type inequalities with a homogeneous kernel involving some differential operators. The main results are proved for the case of non-conjugate parameters. After reduction to the conjugate case, we show that the constants appearing on the right-hand sides of these inequalities are the best possible. MSC:26D10, 26D15, 33B15.

Half-discrete Hilbert-type inequalities with mean operators, the best constants, and applications

In this article we derive several new half-discrete Hilbert-type inequalities with a general homogeneous kernel, involving arithmetic, geometric and harmonic mean operators. The main results are proved for the case of non-conjugate exponents. A special emphasis is given to determining conditions under which these inequalities include the best possible constants. As an application, we consider some operator expressions closely connected to established inequalities.

On Some Multidimensional Hilbert-Type Inequalities in the Discrete Case

Motivated by the results of Huang, we deduce a pair of discrete multidimensional Hilbert-type inequalities involving a homogeneous kernel of negative degree. We also establish conditions under which the constant factors involved in the established inequalities are the best possible. Finally, we consider some particular settings with homogeneous kernels and weight functions. In this way, we obtain generalizations of some results known from the literature.

A STRENGTHENED FORM OF A GENERAL HARDY-TYPE INEQUALITY OBTAINED VIA SUPERQUADRATICITY AND ITS APPLICATIONS

In this paper, we establish a new refined version of the general Hardy-type inequality utilizing the notion of superquadratic functions. The general result obtained is then applied to some particular one-dimensional and multidimensional settings providing generalizations and refinements of some well-known Hardy-type inequalities. Finally, as another consequence of the main result, we also state and prove a class of improved versions of some Hardy- Hilbert-type inequalities.

On several new Hilbert-type inequalities involving means operators

In this paper, we establish several new Hilbert-type inequalities with a homogeneous kernel, involving arithmetic, geometric, and harmonic mean operators in both integral and discrete case. Such inequalities are derived by virtue of some recent results regarding general Hilbert-type inequalities and some well-known classical inequalities. We also prove that the constant factors appearing in established inequalities are the best possible. As an application, we consider some particular settings and compare our results with previously known from the literature.

On more accurate Hilbert-type inequalities in finite measure spaces

In this paper we establish several more precise versions of a general Hilbert-type inequality regarding finite measure spaces. These inequalities are derived by virtue of some new improved versions of the Holder inequality, known from the literature. The main results are then applied to some particular settings with a special emphasis to homogeneous kernels and power weight functions.

Hilbert-type inequalities for Hilbert space operators

In this paper we establish a general form of the Hilbert inequality for positive invertible operators on a Hilbert space. Special emphasis is given to such inequalities with homogeneous kernels. In some general cases the best possible constant factors are also derived. Finally, we obtain the improvement of previously deduced results, based on the application of the Hermite-Hadamard inequality.

A Unified Treatment of Half-Discrete Hilbert-Type Inequalities with a Homogeneous Kernel

In this paper we provide a unified treatment of half-discrete Hilbert-type inequalities with a general homogeneous kernel. The main results are proved for the case of non-conjugate exponents. A special emphasis is given to determining conditions under which these inequalities include the best possible constants. As an application, we consider some operator expressions closely connected to established inequalities. Finally, we also provide improvements of derived half-discrete inequalities by virtue of the Hermite-Hadamard inequality.

Gaussian subordination for the Beurling-Selberg extremal problem

We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function $e^{-\pi\lambda x^2}$ by entire functions of exponential type. This leads to the solution of analogous extremal problems for a wide class of even functions that includes most of the previously known examples (for instance \cite{CV2}, \cite{CV3}, \cite{GV} and \cite{Lit}), plus a variety of new interesting functions such as $|x|^{\alpha}$ for $-1 < \alpha$; \,$\log \,\bigl((x^2 + \alpha^2)/(x^2 + \beta^2)\bigr)$, for $0 \leq \alpha < \beta$;\, $\log\bigl(x^2 + \alpha^2\bigr)$; and $x^{2n} \log x^2$\,, for $n \in \N$. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.

On a multidimensional version of the Hilbert type inequality

The main objective of this paper is a study of some new multidimensional Hilbert type inequalities with a general homogeneous kernel. We derive a pair of equivalent inequalities, and also establish the conditions under which the constant factors included in the obtained inequalities are the best possible. Some applications in particular settings are also considered.

On Hilbert type inequalities

In the present paper we establish new inequalities similar to the extensions of Hilbert’s double-series inequality and also give their integral analogues. Our results provide some new estimates to these types of inequalities.MSC:26D15.

Some Extensions of Hilbert-Type Inequalities

In this paper, an extension of Hilbert-type inequalities with a best constant factor is given by introducing two parameter .

A refined discrete Hilbert inequality obtained via the Hermite–Hadamard inequality

In this paper we establish a general method for improving the discrete Hilbert-type inequalities via the Hermite–Hadamard inequality. The general result is then applied to homogeneous kernels. Moreover, some particular homogeneous kernels are considered, and a whole series of refinements of some recent results known from the literature are yielded. In addition some particular non-homogeneous cases are also considered.

ON SOME REFINEMENTS AND CONVERSES OF MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES

AbstractThe main objective of this paper is a study of some new refinements and converses of multidimensional Hilbert-type inequalities with nonconjugate exponents. Such extensions are deduced with the help of some remarkable improvements of the well-known Hölder inequality. First, we obtain refinements and converses of the general multidimensional Hilbert-type inequality in both quotient and difference form. We then apply the results to homogeneous kernels with negative degree of homogeneity. Finally, we consider some particular settings with homogeneous kernels and weighted functions, and compare our results with those in the literature.

Note on Hilbert-type inequalities

The main objective of this paper is to prove Hilbert-type and Hardy-Hilbert-type inequalities with a general homogeneous kernel, thus generalizing a result obtained in [Namita Das and Srinibas Sahoo, A generalization of multiple Hardy-Hilbert's integral inequality, Journal of Mathematical Inequalities, 3(1), (2009), 139--154].